Name: Conic Sections and Ellipse
Uploaded: 2019-07-19
Duration: 3 min 55 s
Description: In this video, we find the equation of an ellipse that is centred at the origin when the eccentricity and vertices are given. Study this video for a better understanding.

Question

Prove that the middle point of the intercept made on a tangent to a parabola by the tangents at two points

P

and

Q

lies on the tangent which is parallel to

P Q

.

Accepted Answer

Let the parabola be $y^{2} = 4 a x and P (a t_{1}^{2}, 2 a t_{1}) & Q (a t_{2}^{2}, 2 a t_{2})$ be the two points on the parabola.

$Slope of P Q = \frac{2 a t_{2} - 2 a t_{1}}{a t_{2}^{2} - a t_{1}^{2}}$

$= \frac{2 a (t_{2} - t_{1})}{a (t_{2}^{2} - t_{1}^{2})}$

$= \frac{2}{(t_{2} + t_{1})}$

As,we know that the point of intersection of tangent at $P & Q is (a t_{1} t_{2}, a (t_{1} + t_{2}))$ .

Let $R be (a t_{1} t_{2}, a (t_{1} + t_{2}))$ .

Let mid-point of $P R be M_{1}$ .

Coordinates of $M_{1} = (\frac{a t_{1}^{2} + a t_{1} t_{2}}{2}, \frac{a t_{1} + a t_{2} + 2 a t_{1}}{2})$

$= (\frac{a t_{1}^{2} + a t_{1} t_{2}}{2}, \frac{a (3 t_{1} + t_{2})}{2})$

Let mid-point of $Q R be M_{2}$ .

Coordinates of $M_{2} = (\frac{a t_{2}^{2} + a t_{1} t_{2}}{2}, \frac{a t_{1} + a t_{2} + 2 a t_{2}}{2})$

$= (\frac{a t_{2} (t_{1} + t_{2})}{2}, \frac{a (3 t_{2} + t_{1})}{2})$

Slope of $M_{1} M_{2} = \frac{\frac{a (t_{1} + 3 t_{2})}{2} - \frac{a (t_{2} + 3 t_{1})}{2}}{\frac{a t_{2} (t_{1} + t_{2})}{2} - \frac{a t_{1} (t_{1} + t_{2})}{2}}$

$= \frac{\frac{2 (t_{2} - t_{1})}{2}}{\frac{(t_{2} - t_{1}) (t_{1} + t_{2})}{2}}$

$= \frac{2}{(t_{1} + t_{2})}$

Since, slope of $P Q = slope of M_{1} M_{2}$

$∴$ Middle point of intercepts made on a tangent to a parabola by the tangents at $P & Q$ lies on the tangent which is parallel to $P Q$ .

Prove that the middle point of the intercept made on a tangent to a parabola by the tangents at two points $P$ and $Q$ lies on the tangent which is parallel to $P Q$ .

Important Questions on Conic Section

Learn and Prepare for any exam you want !

Prove that the middle point of the intercept made on a tangent to a parabola by the tangents at two points P and Q lies on the tangent which is parallel to PQ.

Important Questions on Conic Section

Learn and Prepare for any exam you want !

Prove that the middle point of the intercept made on a tangent to a parabola by the tangents at two points $P$ and $Q$ lies on the tangent which is parallel to $P Q$ .